eric-talk-1

Genetic Inheritance and Genetic Data

Eric C. Anderson

The Wildlife Society CKMR Workshop, Sunday November 6, 2022

Outline

Motivation
Mendelian Inheritance
Identity-by-descent
Physical linkage
Genetic markers and identity in state
Pairwise joint genotype probabilities
Types of genetic markers

Motivation

Close-kin mark-recapture depends on identifying pairs of related individuals
Genetic data are used to identify these relatives
Relatives can be identified because they “share” more of their genome than unrelated (or less related) individuals.
A rigorous definition of what is meant by “genome sharing” is required.

We look like our relatives due both to shared genetics and shared environment
Quantitative genetics: what fraction of phenotypic resemblance is due to genetics
Kin-finding: using the fraction of shared genetic markers to infer relationship

Starting from the beginning

Gregor Mendel (1822-1884)
Inferred laws of genetic segregation by observing patterns of single-locus traits in pea plants.
Not applicable to most (polygenic) traits
But, we now know his laws are quite applicable to the transmission of genetic material in diploid organisms.
They apply to dicussing the inheritance of a locus.

What the heck is a locus?

The word “locus” in genetics, much like the word “gene” has been applied to a lot of things.
For our purposes, we will use it to refer to a “chunk” of DNA that is defined by its position in the genome.
For example, around base position X on Chromosome 3:

Two copies of the DNA at each locus. One chromosome from mom. Another from dad.

Mendel’s First Law of Segregation (in sexual diploids)

When gametes are formed, the two copies of each locus segregate so that each gamete carries exactly one copy of each locus.

Since individuals are formed by the union of gametes, this dictates the segregation of DNA to offspring:

Each child receives exactly one copy of a locus from its mother and one from its father.
The copy of the locus received from the parent is chosen randomly (with probability \(\frac{1}{2}\) for each) from amongst the two copies in the parent.

Consider the case of the maternal allele segregated to an offspring, in the pedigree to the right.

Circles are females.
Males are squares.

\(\mathrm{Probability} = \frac{1}{2}\)

Mendel’s Second Law

“The gene copy that gets segregated to a gamete/offspring at one locus is independent of the gene copy that gets segregated at another locus.”

This is not universally true.
It is only true for loci that are on different chromosomes.

When two loci are on the same chromosome they are referred to as “physically linked”

AND, they do not segregate independently.

Chromosomes, Crossovers, and Recombination

Chromosomes are inherited in big chunks
Crossovers occur between the maternal and paternal chromosomes of an individual.
An odd number of crossovers between two loci = recombination
Crossover modeled as a Poisson process along chromosomes.
Loci close together are less likely to have a recombination

We will come back to linked loci later today.
For now, we focus on a single locus…

Genetic Identity by Descent

When the DNA at a locus is a direct copy of the DNA in an ancestor…
Or when the stretches of DNA in two different individuals at a locus are both copies of the same piece of DNA in a recent ancestor…
These pieces of DNA are termed identical by descent, (IBD).

Ma and Kid are IBD at one gene copy (fuschia)
Pa and Kid are IBD at one gene copy (blue)

Kid-1 and Kid-2 share 1 gene copy IBD (blue)

Kid-1 and Kid-2 share no gene copies IBD (blue)

The number of gene copies IBD

A pair of non-inbred diploid individuals can share 0, 1, or 2 gene copies IBD at a locus.
Each possible relationship between two non-inbred individuals can be characterized by the expected fraction of the genome at which the two individuals share 0, 1, or 2 gene copies IBD. \[ \boldsymbol{\kappa}= (\kappa_0, \kappa_1, \kappa_2) \]
- \(\kappa_0\): expected fraction of genome with 0 gene copies IBD
- \(\kappa_1\): expected fraction of genome with 1 gene copy IBD
- \(\kappa_2\): expected fraction of genome with 2 gene copies IBD
These are also the marginal probabilities that a pair of individuals share 0, 1, or 2, gene copies at a single locus.
Sometimes called “Cotterman coefficients”, we will call them “kappas”
These are not the coefficient of relationship or coefficient of consanguinity. (Those are not sufficient).

\(\boldsymbol{\kappa}\) for Parent-Offspring Pairs

We start with an easy one
What is \(\boldsymbol{\kappa}\) for the parent offspring relationship?
Well, a parent and offspring always share exactly one gene copy IBD.
\[ \begin{aligned} \kappa_0 &= 0 \\ \kappa_1 &= 1 \\ \kappa_2 &= 0 \\ \end{aligned} \]
or \[ \boldsymbol{\kappa}= (0, 1, 0) \]

\(\boldsymbol{\kappa}\) for Half-Sibling Pairs

A slightly harder one:
We know that, with probability 1, Pa segregates a copy of a single gene to Kid-1
With probability \(\frac{1}{2}\) Pa segregates of a copy of the same gene to Kid-2
With probability \(\frac{1}{2}\) Pa segregates a copy of the other gene to Kid-2
\[ \begin{aligned} \kappa_0 &= 1/2\\ \kappa_1 &= 1/2 \\ \kappa_2 &= 0 \\ \end{aligned} \]

\(\boldsymbol{\kappa}\) for Full-Sibling Pairs

Like two independent half-sib relationships…
With probability \(\frac{1}{2}\), Kid-1 and Kid-2 share 1 gene copy (or 0 gene copies) IBD from Ma
Independently, with probability \(\frac{1}{2}\), Kid-1 and Kid-2 share 1 gene copy (or 0 gene copies) IBD from Pa.
So, no gene copies IBD = none from Ma and none from Pa = \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\).
Two gene copies IBD = one from Ma and one from Pa = \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\)
One gene copy IBD = one from Ma and zero from Pa, or zero from Ma and one from Pa = \(\frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} = \frac{1}{2}\).
So, \(\kappa_0 = \frac{1}{4}~~~~~~\kappa_1 = \frac{1}{2}~~~~~~\kappa_1 = \frac{1}{4}\)